A Card Game for Bell's Theorem and Its Loopholes Guy Vandegrift¹, Joshua Stomel²

WikiJournal of Science, 2018, 1(1):6 doi: 10.15347/wjs/2018.005 Research Article

A card game for Bell's theorem and its loopholes Guy Vandegrift¹, Joshua Stomel²

Abstract In 1964 John Stewart Bell made an observation about the behavior of particles separated by macroscopic distances that had puzzled physicists for at least 29 years, when Einstein, Podolsky and Rosen put forth the famous EPR paradox. Bell made certain assumptions leading to an inequality that entangled particles are routinely observed to violate in what are now called Bell test experiments. As an alternative to showing students a "proof" of Bell's inequality, we introduce a card game that is impossible to win. The solitaire version is so simple it can be used to introduce binomial statistics without mentioning physics or Bell's theorem. Things get interesting in the partners' version of the game because Alice and Bob can win, but only if they cheat. We have identified three cheats, and each corresponds to a Bell's theorem "loophole". This gives the instructor an excuse to discuss detector error, causality, and why there is a maximum speed at which information can travel.

The conundrum any theory that might replace quantum mechanics.[6] It is also possible to view a loophole as a physical mecha- Although this can be called a theorem, it might be bet- nism by which the outcome of a Bell's theorem experi- ter viewed as something "spooky" that has been rou- ment might seem less "spooky". In this paper, we asso- tinely observed, and is consistent with quantum me- ciate loopholes with ways to cheat at the partners' ver- chanics. But this puzzling behaviour violates what sion of the card game. It should be noted that the might be called common notions about what is and is three loophole mechanisms introduced in this paper not possible.[1][2] Students typically encounter a math- raise questions that are even spookier than quantum ematical theorem as an incomprehensible statement mechanics: Are the photons "communicating" with each that cannot be digested until it is first proven and then other? Do they "know" the future? Do they "persuade" applied in practice. It is not uncommon for novices to the measuring devices to fail when the "cards are unfa- refer to some version of Bell's inequality as Bell's theo- vorable"?[7] rem because the inequality can be mathematically "proven".[3] The problem is that what is proven turns out Since entanglement is so successfully modeled by to be untrue. quantum mechanics, one can argue that there is no need for a mechanism that "explains" it. Nevertheless, David Mermin described an imaginary device not unlike there are reasons for investigating loopholes. At the that shown in Fig. 1, and refers to the fact that such a most fundamental level, history shows that a successful device actually exists as a conundrum, then pointed out physical theory can be later shown to be an approxima- that many physicists deny that it is a conundrum.[4] tion to a deeper theory, and the need for this new theory is typically associated with a failure of the old para- A simple Bell's theorem experiment digm. It is plausible that a breakdown of quantum me- It is customary to name the particles[5] in a Bell's theo- chanics might be discovered using a Bell's theorem ex- rem experiment "Alice" and "Bob", an anthropomor- periment designed to investigate a loophole. But the phism that serves to emphasize the fact that a pair of vast majority of us (including most working physicists) humans cannot win the card game ... unless they need other reasons to care about loopholes: Many find cheat. To some experts, a "loophole" is a constraint on it interesting that we seem to live in a universe gov- erned by fundamental laws, and Bell's theorem yields insights into the bizarre nature of those laws. Also, 1 Wright State University Lake Campus, Celina, Ohio, 45822 those who teach can use these card games to motivate 2 Wright State University Lake Campus, Celina, Ohio, 45822 *Author correspondence: [email protected] ORCID: [0000-0000-0000-0001] Supplementary material: commons.wikimedia.org/location Licensed under: CC-BY Received 02-02-2018; accepted 31-05-2018

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introductory discussions about statistical inference, po- while the odd cards (3,5,7,9) represent a photon being larization, and modern physics. blocked.

Although Bell's inequality is easy to prove[9], we avoid it here because the card game reverses roles regarding probability: Instead of the investigators attempting to ascertain the photons' so-called hidden variables, the players are acting as particles attempting to win the game by guessing the measurement angles. Another Figure 1 | The outside casing of each device remains station- complication is that the original form of Bell's inequal- ary while the circle with parallel lines rotates with the center ity does not adequately model the partners' version of arrow pointing in one of three directions (♥, ♣, ♠.) If Jacks the game because humans have the freedom to exhibit are used to represent these directions, Alice will see J♥ as her question card. She will respond with an "odd"-numbered a behavior not observed by entangled particles (under answer card (3♥) to indicate that she is blocked by the filter. ideal experimental conditions). This behavior involves a If Bob passes through a filter with the "spade" orientation, 100% correlation (or anti-correlation) whenever the po- he sees J♠ as the question card, and answers with the "even" larization measurement angles are either parallel or numbered 2♠. This wins one point for the team because perpendicular to each other.[10] In the partners' version they gave different answers to different questions. of the card game, this behavior must be enforced by de- ducting a penalty from the partners' score whenever Figure 1 shows a hypothetical and idealized experi- they are caught using a forbidden strategy (which we ment involving two entangled photons simultaneously shall later call the β-strategy). The minimum required emitted by a single (parent) atom. After the photons penalty is calculated in Supplementary file:The car have been separated by some distance, each is ex- and the goats. Fortunately students need not master posed to a measurement that determines whether the this calculation because the actual penalty should often photon would pass or be blocked by the polarizing fil- be whatever it takes to encourage a strategy that mim- ter.[8] To ensure that the results seem "spooky" it ics this aspect of entanglement (which we shall call the should be possible to rotate the filter while the pho- α-strategy.) tons are en route so that the filter's angle of orienta- A theoretical understanding of how one can model en- tion is not "known" to either photon until it encounters tanglement using the Schrödinger equation can be the filter. If the filters are rotated between only three found in Supplementary file:Tube entanglement. polarization angles, we may use card suits (hearts ♥, clubs ♣, spades ♠) to represent these angles. These The solitaire card game three polarization angles are associated with "ques- Figure 2 shows the three possible outcomes associated tion" cards, because the measurement essentially asks with one hand of the solitaire version of the game. The the photon a question: solitaire version requires nine cards. The figure uses a set with three "jacks" (♥ ♣ ♠) for the questions, and (2,3) "Will you pass through a filter oriented at for the six (even/odd) answer cards. To play one round this angle?" of the game, the player first shuffles the three question For simplicity we restrict our discussion to symmetric cards and places them face down so their identity is not angles (0°, 120°, 240°.) The filter's axis of polarization known. Next, for each of the three suits, the player se- is shown in the figure as parallel lines, with the center lects an even or odd answer card. The figure shows the line pointing to the heart, club, or spade. Any face card player choosing the heart and club to be even, while the can be used to "ask" the question, and the four face spade is odd: 2♥ 2♣ 3♠. This is the only viable strategy, cards (jack, queen, king, ace) are equivalent. If the de- since the alternative is to always lose by selecting three tectors are flawless, each measurement is binary: The answers that are all even or all odd. In the partner's ver- photon either passes or is blocked by the filter (subse- sion we shall introduce a second, β-strategy, which is quent measurements on a photon would yield nothing not possible in the solitaire game. interesting.) The measurement's outcome is repre- After three answer cards are selected and turned face sented by an even or odd numbered "answer" card (of up, two of the three question cards are randomly se- the same suit). The numerical value of an answer card lected and also turned face up. Figure 2 depicts all three is not important: all even numbers (2,4,6,8) are equiva- equally probable outcomes, or ways to select two out of lent and represent a photon passing through the filter,

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three cards (3 choose 2.)[11] The round is scored by add- with a probability of 3/4. Table 1 shows that this scoring ing or subtracting points, as shown in Table 1: First the system causes humans to average a loss of at least 1/3 suit of each of the two upturned question cards is of a point per round, while entangled particles maintain matched to the corresponding answer card. In case 1 an average score of zero.[12] How do particles succeed (shown in the figure), the player wins one point because where humans fail? answers are different: ♥ is an even number, while ♠ is Points Answers are: [13] odd. The player loses three points in case 2 because Example +1 different ♥ ♠ the ♥ and ♣ are the same (even). Case 3 wins one point 2 and 3 +1 different ♥ ♣ for the player because the answers are different. It is ev- 2 and 3 −3 same ♥ ♣ ident that the player has a 2/3 probability winning a 2 and 2 round. The conundrum of Bell's theorem is that entan- Table 1 | Solitaire scoring gled particles in an actual experiment manage to win

- Figure 2 | Solitaire version of game. Cases 1, 2, and 3 represent three possible outcomes if the player chooses the best strategy (later called the "α-strategy": One answer (here, "odd" for ♠) differs from that given for the other two questions (here, "even" for ♥ & ♣.) The game for entangled partners is impossible. In this phase each player silently plays an (even/odd) answer that matches the question's suit. In the partners' version, Alice and Bob each play one The player cannot know the other's question or answer (even/odd) answer card in response to the suit of a ques- during phase 2. tion card. Every round is played in two distinctly differ- In the solitaire version, the player held a deck of six ent phases. Alice and Bob are allowed to discuss strat- numbered cards and pre-selected (even/odd) answers egy during phase 1 because it simulates the fact that the for each of the three (question) suits. This simulated the particles are (effectively) "inside" the parent atom be- parent atom "deciding" the responses that each photon fore it emits photons. Then, all communication be- will give to all possible polarization measure- tween the partners must cease during phase 2, which [14] simulates the arrival of the photons at the detectors for ments. In an "ideal" Bell's theorem experiment, the measurement under conditions where communication two photons' responses to identical polarization meas-

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urement angles are either perfectly correlated or per- 4 1−푃푆 푄 ≥ ( ), (1) fectly anticorrelated.[8][15] This freedom to inde- 3 푃푆 pendently choose different answers when Alice and where PS is the probability that Alice and Bob are asked Bob are faced with the same question creates a di- the same question. The equality holds if PS = 1⁄4 and lemma for the designers of the partners' version of the 푄 = 4, which can be accomplished by randomly select- card game. Adherence to any rule forbidding different ing two question cards from nine (K♠, K♥, K♣, Q♠, Q♥, answers to the same question cannot always be veri- Q♣,J♠, J♥, J♣), as shown in Fig. 3. If the equality in (1) fied. To enforce this rule, we deduct 푄 points whenever holds, the partners are "neutral" with respect to the se- they give different answers to the same question. No lection of two different strategies, one of which risks points are awarded for giving the same answer to the the 4 point penalty. Both strategies lose, but the loss same question. Note how this complexity is relevant to rate is reduced to −1/4 points per round, because the actual experiments because detectors can register false referee must dilute the number of times different ques- events. The minimum penalty that should be imposed tions are asked depends on how often the partners are given question cards of the same suit, and is derived at Supplementary file:The car and the goats

Figure 3 | One round of the partners' version with Alice and Bob employing the same strategy (α) introduced in the solitaire game. Here, a version of "neutral" scoring is used in which the referee randomly selects from the nine question cards, with a penalty of 4 points assessed if different answers are given to the same question. Instructors might wish to override this "neutral" scoring by asking the same question more often than called for in the random selection

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A sample round begins in the top part of Fig. 3 as phase use of the α-strategy as "biased scoring". To further in- 1 where the pipe smoking referee has selected different hibit use of the β-strategy, the referee should routinely questions (hearts and spades). In a classroom setting, override the shuffle and deliberately select question consider allowing Alice and Bob to side-by-side, facing cards of the same suit. The distinction between biased slightly away from each other during phase 2. Arrange and neutral scoring lies in whether the equality or the for the audience to sit close enough to listen and watch inequality holds in (1). Table 2 shows examples of each for evidence of surreptitious communication between scoring system. Both were selected to match an integer Alice and Bob. The prospect of cheating not only makes value for 푄. The shuffle of 9 face cards exactly matches the game more fun, but also allows us to introduce the equality in (1) if 푄 = 4, while the more convenient "loopholes". The "thought-bubbles" above the partners collection of 6 face cards will bias the players towards show a tentative agreement by the partners to play the the the α-strategy if 푄 = 6[16] same α-strategy introduced in the solitaire version Shuffle 9 face cards to ask Neutral scoring: (both say "even" to ♥♣, and "odd" to ♠.) It is important the same question exactly Points if Bob and Alice give… to allow both players to hold all the answer cards in 25% of the time. phase 2 so that each can change his or her mind upon +1 different answers to "even" to hearts seeing the actual question. The figure shows them fol- different questions "odd" to spades lowing their original plan and winning because the ref- −3 the same answer to "even" to clubs eree selected a heart for Alice and a spade for Bob. different questions "even" to hearts −4 Different answers to "even" to clubs But the partners have another strategy that might win: the same question "odd" to clubs Suppose Alice agrees to answer "even" to any ques- 0 the same answer to "even" to clubs tion, while Bob answer is always "odd". This wins (+1) if the same question (for both players) different questions are asked, and loses (−Q) if the same question is asked. This is called the β-strategy. Shuffle 6 face cards The Supplementary file:The car and the goats estab- Biased scoring: and/or ask the same ques- lishes that no other strategy is superior to the α and/or Points if Bob and Alice give… tion with a probability β strategies: higher than 2/11. +1 different answers to "even" to hearts α-strategy: Alice and Bob select their answers in ad- different questions "odd" to spades vance, in such a way that both give the same answer −3 the same answer to "even" to clubs if asked the same question. For example, they might different questions "even" to hearts both agree that ♥♣ are even, while ♠ is odd. This −4 Different answers to "even" to clubs strategy was ensured in the solitaire version because the same question "odd" to clubs only three cards are played: If the heart is chosen to 0 the same answer to "even" to clubs the same question (for both players) be "even", the solitaire version models a situation where both Alice and Bob would answer "even" to Table 2 | Examples of neutral and biased scoring "heart". This α-strategy requires that one answer differs from the other two (i.e., all "even" or all "odd" is never a good strategy). The expected loss is 1/3 for Cheating at cards and Bell's theorem each round whenever different questions are asked. "loopholes" β-strategy: One partner always answers "even" while the other always answers "odd". This strategy In the card game, Alice and Bob could either win by sur- gains one point if different questions are asked, and reptitiously communicating after they see their ques- loses 푄 points if the same question is asked. tion cards, or by colluding with the referee to learn the questions in advance. Which seems more plausible, in- For pedagogical reasons, the instructor may wish to dis- formation travelling faster than light, or atoms acting as courage the β-strategy. If Alice and Bob are not asked if they "know" the future? A small poll of undergraduate the same question often, they might choose to risk math and science college students suggests that they large losses for the possibility winning just a few rounds are inclined to favor faster-than-light communica- using the β-strategy, perhaps terminating the game tion as being more plausible. We shall use a space-time prematurely with a claim that they lost "quantum en- diagram to illustrate how faster-than-light communica- tanglement". To counter this, the referee can raise the tion violates causality by allowing people to send sig- penalty to six points and randomly shuffle only six ques- nals to their own past. And, we shall argue that one can tion cards that result from the merging of two solitaire decks. We refer to any scoring that favors the players'

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make the case that decisions made today by humans re- upon seeing his or her question card. After convincing garding how and where to perform a Bell's theorem ex- ourselves that no superior strategy exists, we realized periment next week, might be mysteriously connected that a player could cheat by terminating the game after to the behavior of an obscure atom in a distant galaxy seeing his or her own question card, but before playing billions of years ago.[17] the answer card. This is related to an important detector efficiency loophole.[19] The student's discovery also The third loophole was a surprise for us. In an early trial [18] alerted us to the fact that our original calculation of (1) of the partners' game, a student stopped playing was just a lucky guess based on flawed logic. and attempted to construct a modified version of the α- strategy that uses the new information a player gains

Referee collusion: Determinism loophole Alice and Bob could win every round of the partners' version if they cheat by communicating with each other after seeing their question cards in phase 2. In an actual experiment, this loophole is closed by making the measurements far apart in space and nearly simultaneous, which in effect requires that these communications travel faster than the speed of light.[20] While any faster-than-light communication is inconsistent with special relativity, we shall limit our discussion to information that travels at nearly infinite speed.[21] Figure 4| "Magic phone#1" is situated on a moving train and Figure 4 shows Alice and Bob slightly more than one can be used by Alice to send a message to Bob's past, which light-year apart. The dotted world lines for each is ver- Bob relays back to Alice's past using the land-based "Magic tical, indicating that they remain at rest for over a year. phone #2". These magic phones transmit information with The slopes of world lines of the train's front and rear are near infinite speed. roughly 3 years per light-year, corresponding to about 1/3 the speed of light. Both train images are a bit con- The slanted image of the train depicts the location of fusing because it is difficult to represent a moving train each car on the day that the (moving) passengers per- on a space-time diagram: A moving train can be defined ceive the front to be adjacent to Alice, at the same time by the location of each end at any given instant in time. that the train's rear is perceived to be adjacent to Bob. This requires the concept of simultaneity, which is per- It should be noted that Alice and Bob do not perceive ceived differently in another reference frame. The hori- these two events as simultaneous. The figure shows zontal image of the train at the bottom represents to that the rear passes Bob several months before the location of each car on the train on the first day of Jan- front passes Alice (in the partners' reference frame.) uary, as time and simultaneity are perceived by Alice Now we establish that the passengers perceive the and Bob. To complicate matters, the horizontal train front of the train to reach Alice at the same time that image is not what they would actually see due to the fi- the rear reaches Bob. The light-emitting-diode (LED) nite transit time required for light to reach their eyes. It shown at the bottom of Fig. 4 emits two pulses from the helps to imagine a distant observer situated on a per- center of the train in January. It is irrelevant whether the pendicular to some point on the train. The transit time LED is stationary or moving because all observers will for light to reach this distant observer will be nearly the see the pulses travelling in opposite directions at the same for every car on the train. Many years later, this speed of light (±1 ly/yr.) Note how the backward mov- distant observer will see the horizontal train as depicted ing pulses reaches the rear of the train in May, five at the bottom of the figure. It will be instructive to re- months before the other pulse reaches the train's front turn to the perspective of this distant observer after the in October. But, the passengers see two light pulses cre- paradox has been constructed. ated at the center of the train, directed at each end of the train, and will therefore perceive the two pulses as striking simultaneously.

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To create the causality paradox, we require two "magic- the detectors as one integrated quantum entity is con- phones" capable of sending messages with nearly infi- sistent with the proper modeling of a quantum-me- nite speed. Unicorn icons use arrows to depict the infor- chanical system. The paradoxical violation of Bell's ine- mation's direction of travel: magic phone #1 transmits quality arises from the need to model two remote par- from Alice to Bob, while #2 transmits from Bob to Alice. ticles as one system, so it is not unreasonable to as- Magic phone #1 is situated on the moving train. When sume that the conundrum can be resolved by including Alice shows her message through the front window as the devices that make the measurements into that the train passes her in October, a passenger inside re- model. lays the message via magic phone #1 to the train's rear, where Bob can see it through a window. Bob immediately relays the message back to Alice via the land- based magic phone #2 in May, five months before she sent it. Our distant observer will likely take a skeptical view of all this. The slope of the slanted train's image indicates that the distant observer will see magic phone #1 sending information from Bob to Alice, opposite to what the passengers perceive. The distant observer will first see the message inside the rear of the train (when it was adjacent to Bob in May). That message will immediately Figure 5 | Cosmic photons from two distant spiral galaxies ar- begin to travel towards of Alice, faster than the speed rive on Earth with properties that trigger the filters to ask the of light, but slow enough so that Alice will not receive it ♥ & ♠ questions of photons just prior to their arrival with a until October. Meanwhile, Bob sends the same mes- winning combination of (even/odd) answers. sage via land-based phone #2 to Alice, who receives it in May. Alice waits for almost five months, until she pre- Figure 5 is inspired by a comment made by Bell during pares to send the same message, showing it through a 1985 radio interview that mentioned something he the front window just before the message also arrives called "superdeterminism".[22][23] It is a timeline that at the front via the train-based magic phone #1. It depicts the big bang, beginning at a time when space would appear to the distant observer that the events and time were too confusing for us to graph. At this depicted in Fig. 4 had been artificially staged. beginning, "instructions" were established that would This communications loophole in an actual Bell test ex- dictate the entire future of the universe, from every ac- periment was closed by arranging for the measure- tion taken bywevery human being, to the energy, ments to coincide so that any successful effort to com- path, and polarization of every photon that will ever municate would suggest that humans could change exist. Long ago, obscure atoms in two distant galaxies their own past using this ability to send information (Sb and Sc) were instructed to each emit what will be- faster than light. come "cosmic photons" that strike Earth. Meanwhile, "instructions" will call for humans to evolve on Earth Referee collusion: Determinism and create a Bell's theorem experiment that uses the frequency and/or polarization of cosmic photons to set loophole the polarization measurement angles while the entan- This "determinism", or "freedom-of-choice" loophole in- gled photons Alice and Bob are still en route to the de- volves the ability of the quantum system to predict the tectors. Alice and Bob will arrive at their destinations future. Curiously, the strategy would not be called already "knowing" how to respond because the cosmic "cheating" in the card game if Alice or Bob relied on in- photons were "instructed" to have properties that tuition to guess which cards the referee will play in the cause the questions to be "heart" and "spade". upcoming round. But what makes this loophole bizarre Viewed this way, the events depicted in Fig. 5 are just when applied to a Bell test experiment is that it would the way things happen to turn out. Efforts to enact the have been necessary to predict the circumstances un- scenario with an actual experiment using cosmic pho- der which the experiment was designed and con- tons in this way are being carried out. The most recent structed by human beings who evolved on a plant that experiment looks back at photons created 600 years was formed almost five billion years ago. On the other ago.[24][25] Note also how this experiment does not hand, viewing the parent atom, the two photons, and "close" the loophole, but instead greatly expands the

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scale of any "collusion" between the parent atom and But their chances of winning are reduced to only 50% if detectors. Bob sees the "undesired" club or spade. To avoid raising suspicion, Bob does not stop the game each time he It is claimed that the results of Bell test experiments do sees an unfavorable question. Instead, he stops with a not contradict special relativity, despite what may ap- 50% probability upon seeing an unfavorable card. To pear to some as faster-than-light "communication" be- [26] calculate the average score, we construct a probability tween Alice and Bob. Figure 5 can help us visualize space consisting of equally probable outcomes, begin- this if the "instructions" represent the time evolution of ning with the three possible suits that Bob might see. an exotic version of Schrödinger's equation for the en- We quadruple the size of this probability space (from 3 tire universe. If this wave equation is deterministic, fu- to 12) by treating the following two pairs of events as ture evolution of all probability amplitudes is predeter- independent, and occurring with equal probability: mined. One flaw in this argument is that it relies on an equation that governs the entire universe, and for that 1. Bob will either stop the hand, or play round (Do reason is not likely to be solved or written down. Per- stop or Don't stop.) haps this is why the paradox seems to have no satisfac- 2. After seeing his question, Bob knows that Alice tory resolution. might receive one of only two possible questions (ignoring rounds with the same question asked of both.) The Rimstock cheat: Detector error loophole The following variation of the α-strategy allows the team to match the performance of entangled particles by achieving an average score of zero: Alice preselects three answers and informs Bob of her decision. Bob will either answer in the same fashion, or he might abruptly stop the hand upon seeing his question card, perhaps requesting that the team take a brief break while another pair of students play the role of Alice and Bob. In a card game, this request to stop and replay a hand would require the cooperation of a gullible scorekeeper.

But no detector in an actual Bell's theorem experiment is 100% efficient, and this complicates the analysis of a Figure 6 can be used to show that Bob will stop the [27] Bell's theorem experiment in a way that requires both game with a probability of 1/3. But if Bob and Alice careful calibration of the detector's efficiency, as well as randomly share this role of stopping the game, each detailed mathematical analysis. player will stop a given round with only 1/6, yielding an apparent detector efficiency of 5/6 = 83.3%.[19] Typical results for a team playing this ruse are illustrated in Fig. 7. Ten rounds are played on four different occasions. The vertical axis represents in the team's net score, with upward steps corresponding to winning one point, and downward corresponding to losing three Figure 6 | The Rimstock cheat: Bob flips a coin to determine points. The horizontal lines showing no change in score whether to play the cheat on this round. Alice will play "even" indicate occasions where Bob or Alice refused to play an to hearts and "odd" to spades or clubs. answer card (it was never necessary to ask both part- Since this strategy never calls for Alice and Bob to give ners the same question in this simulation.) different (even/odd) answers to the same question, we Figure 6 was generated using an Excel spreadsheet us- may consider only rounds where the players get differ- ing the rand() function, which caused the graphs to ent questions. To understand why Bob might refuse to change at every ctrl+= keystroke. It took several strokes play a card, suppose Alice plans to answer "even" to to get a graph where the lines did not cross, and all the hearts and "odd" to clubs and spades. As indicated at event counts were this close to expected values. As dis- the top of Fig. 6, Bob the heart is the "desired" suit be- cussed in a supplement, an Excel verification lab is an cause he knows they will win if he sees that question. appropriate activity in a wide variety of STEM courses.

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Supplementary material Subpages

Pedagogical issues Supplementary file 1 | The car and the goats ( Down- load) A rigorous proof of the penalty that yields "neu- To make sixteen solitaire decks, purchase three identi- tral scoring". cal standard 52 card decks. Remove only one suit (hearts, clubs, spades) from each deck to create four Supplementary file 2 | Impossible correlations ( solitaire sets. Each group should contain 3-5 people, Download) Extends Bell's inequality to non-symmetric and two solitaire decks (for "biased" scoring with Q=6.) cases and also proves the CHSH inequality without us- To avoid confusion of an ace (question card) with an ing calculus. (even/odd) answer card, reserve the ace for groups with large even/odd number cards. For example, one group Supplementary file 3 | Tube entanglement ( Down- might have solitaire sets with (ace,8,9) and (king, 6,7). load) Describes a simple analog to entanglement with In a small classroom, the entire audience can observe or polarized photons. It rely's on Maluss' Law, and also in- even give advice to one pair playing the partners' ver- troduces Dirac notation as a shorthand representation sion at the front of the room. Placing the question cards for the wavefunction of two non-interacting massive adjacent to the players at the start will permit the in- particles confined to a narrow tube. structor and entire class to join the partners' discussion regarding strategy during phase 1. For "neutral" scoring with Q=4, the instructor can either borrow question Acknowledgements cards from the class, or convert unused "10" cards into Versions of this manuscript have received five referee questions. Since cheating will come so naturally, this reports (it was first submitted to the American Journal game is not suitable for gambling (even for pennies). of Physics.) It is obvious that each referee was highly Bell's theorem can lead to topics ranging from baseless qualified, and that each exerted a considerable effort pseudoscience to legitimate (but pedagogically unnec- to improve the quality of this paper. essary) speculation regarding alternatives to the theory of quantum mechanics. While few physicists are experts in such topics, all teachers will eventually face such is- Competing interests sues in the classroom. The authors of this paper claim Guy Vandegrift is a member of the WikiJournal of Sci- no expertise in any of this, and the intent is to illustrate ence editorial board. the "spookiness" of Bell's theorem, show how one can use simple logic to prove that faster-than-light communication violates special relativity,[21] and introduce stu- References) dents to the concept of a "deterministic" theory or model.[26] 1. ↑ Bell, John S. (1964). "On the Einstein Podolsky Rosen Paradox". Physics 1 (3): 195–200. doi:10.1103/physicsphysiquefizika.1.195. https://wikiversity.miraheze.org/wiki/File:Bell_1964_on_EPR_paradox.pdf. 2. ↑ Vandegrift, Guy (1995). "Bell's theorem and psychic phenomena". The Philosophical Quarterly 45 (181): 471–476. doi:10.2307/2220310. http://www.wright.edu/~guy.vandegrift/shortCV/Papers/bell.pdf. 3. ↑ See for example this discussion on the Wikipedia article's talk page, or Wikipedia's effort to clarify this with w:No-go theorem 4. ↑ Mermin, N. David (1981). "Bringing home the atomic world: Quantum mysteries for anybody". American Journal of Physics 49 (10): 940–943. doi:10.1119/1.12594. https://wikiversity.miraheze.org/wiki/File:Mer- min_AJP_mysteries_for_everybody_1981.pdf. "(referring to those who do not consider this a conundrum) In one sense they are obviously right. Compare Tovey's remark that (Beethoven's) Waldstein Sonata has no more business than sunsets and sunrises to be paradoxical." 5. ↑ or detectors 6. ↑ These (hypothetical) theories are called "hidden variable" theories Lars- son, Jan-Åke (2014). "Loopholes in Bell inequality tests of local realism". Journal of Physics A: Mathematical and Theoretical 47 (42): 424003. doi:10.1088/1751-8113/47/42/424003. 7. ↑ In w:special:permalink/829073568 these questions are associated with the "communication (locality)", the "free choice" and a "fair sampling" loophole, respectively. 8. ↑ 8.0 8.1 In most experiments electro-optical modulators are used instead of polarizing filters, and often it is necessary to rotate one set of orientations by 90°. Giustina, Marissa; Versteegh, Marijn A. M.; Wengerowsky, Sören;

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Handsteiner, Johannes; Hochrainer, Armin; Phelan, Kevin; Steinlechner, 19. ↑ 19.0 19.1 Garg, Anupam; Mermin, N. David (1987). "Detector inefficiencies Fabian; Kofler, Johannes et al. (2015). "Significant-Loophole-Free Test of in the Einstein-Podolsky-Rosen experiment". Physical Review D 35 (12): Bell’s Theorem with Entangled Photons". Physical Review Letters 115 (25): 3831–3835. doi:10.1103/physrevd.35.3831. 250401. doi:10.1103/physrevlett.115.250401. 20. ↑ Aspect, Alain; Dalibard, Jean; Roger, Gérard (1982). "Experimental Test 9. ↑ Maccone, Lorenzo (2013). "A simple proof of Bell's inequality". American of Bell's Inequalities Using Time- Varying Analyzers". Physical Review Let- Journal of Physics 81 (11): 854–859. doi:10.1119/1.4823600. ters 49 (25): 1804–1807. doi:10.1103/physrevlett.49.1804. 10. ↑ See equation 29 in Aspect, Alain (2002). "Bell's theorem: the naive view 21. ↑ 21.0 21.1 Liberati, Stefano; Sonego, Sebastiano; Visser, Matt (2002). of an experimentalist". In Bertlmann, Reinhold A.; Zeilinger, Anton. Quan- "Faster-than-c Signals, Special Relativity, and Causality". Annals of Phys- tum [un] speakables (PDF). Berlin: Springer. p. 119-153. doi:10.1007/978-3- ics 298 (1): 167–185. doi:10.1006/aphy.2002.6233. 662-05032-3_9. 22. ↑ Bell, John S. (2004). "Introduction to hidden-variable question". Speaka- 푛 ↑ ( ) ble and unspeakable in quantum mechanics: Collected papers on quantum 11. 푘 or "n choose k" is defined in w:Binomial coefficient 12. ↑ The player can lose more than 1/3 of a point per round by adopting the philosophy. Cambridge University Press. pp. 29–39. obviously bad strategy of making all three answers the same (all even or doi:10.1017/cbo9780511815676.006. all odd.) This is closely related to the fact that Bell's "inequality" is not 23. ↑ Kleppe, A. (2011). "Fundamental Nonlocality: What Comes Beyond the Bell's "equation". Standard Models". Bled Workshops in Physics. 12. pp. 103–111. In that in- 13. ↑ Since 3-choose-2 equals 6, three other cases exist; all involve 3♥. terview, Bell was apparently speculating about a deterministic "hidden 14. ↑ Keep in mind that it seems artificial for the parent atom to "know" that variable theory where all outcomes are highly dependent on initial condi- these photons are part of an experiment involving just three possible po- tions. larization measurements. This need to somehow orchestrate all possible 24. ↑ Gallicchio, Jason; Friedman, Andrew S.; Kaiser, David I. (2014). "Testing fates for each emitted photon created the EPR conundrum long before Bell’s Inequality with Cosmic Photons: Closing the Setting-Independence Bell's inequality was discovered. See w:EPR paradox. Loophole". Physical Review Letters 112 (11): 110405. 15. ↑ It is best not to assume that this correlations implies that the "decision" doi:10.1103/physrevlett.112.110405. regarding polarization was actually made as the two photons are created 25. ↑ Handsteiner, Johannes; Friedman, Andrew S.; Rauch, Dominik; Gallic- by the parent atom. In physics, mathematical models should be judged by chio, Jason; Liu, Bo; Hosp, Hannes; Kofler, Johannes; Bricher, David et al. whether they yield predictions that can be verified by experiment, not (2017). "Cosmic Bell Test: Measurement Settings from Milky Way Stars". whether these models make any sense. Physical Review Letters 118 (6): 060401. 16. ↑ Equation (1) shows that the case is neutral at doi:10.1103/PhysRevLett.118.060401. 26.0 26.1 17. ↑ One can also make the case that it is not the role of physics (or science) 26. ↑ See also Ballentine, Leslie E.; Jarrett, Jon P. (1987). "Bell's theo- to speculate in such matters. rem: Does quantum mechanics contradict relativity?". American Journal of 18. ↑ User:Rimstock Physics 55 (8): 696–701. doi:10.1119/1.15059. 27. ↑ 2/3 is the product of the probability of receiving an unfavorable card, and 1/2 is the probability of stopping; hence (2/3)(1/2)=1/3

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